A One Step Method for the Solution of General Second Order Ordinary Differential Equations

In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better

Modified Block Method for the Direct Solution of Second Order Ordinary Differential Equations

The direct solution of general second order ordinary differential equations is considered in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified the existing block method in order to accommodate the general nth order ordinary differential equation. The method was found to be efficient when tested on second order ordinary differential equation

Numerical algorithm of block method for general second order ODEs using variable step size

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm

Programming codes of block-Milne's device for solving fourth-order ODEs

Block-Milne’s device is an extension of block-predictor-corrector method and specifically developed to design a worthy step size, resolve the convergence criteria and maximize error. In this study, programming codes of block- Milne’s device (P-CB-MD) for solving fourth order ODEs are considered. Collocation and interpolation with power series as the basic solution are used to devise P-CB-MD. Analysing the P-CB-MD will give rise to the principal local truncation error (PLTE) after determining the order. The P-CB-MD for solving fourth order ODEs is written using Mathematica which can be utilized to evaluate and produce the mathematical results. The P-CB-MD is very useful to demonstrate speed, efficiency and accuracy compare to manual computation applied. Some selected problems were solved and compared with existing methods. This was made realizable with the support of the named computational benefit

One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations

Real life problems particularly in sciences and engineering can be expressed in differential equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally solved by first converting them into their equivalent systems of first order ordinary differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently, hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step hybrid block methods with generalised off-step point(s) for solving higher order ordinary differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential equations directly

Predictor-corrector scheme in modified block method for solving delay differential equations with constant lag

In this paper, the numerical solution of delay differential equations using a predictor-corrector scheme in modified block method is presented. In this developed algorithm, each coefficient in the predictor and corrector formula are recalculated when the step size changing. The Runge-Kutta Fehlberg step size strategy has been applied in the algorithm in order to achieve better results in terms of accuracy and total steps. Numerical results are given to illustrate the performance of this modified block method for solving delay differential equations with constant lag

A Computational Approach in Estimating the Amount of Pond Pollution and Determining the Long Time Behavioural Representation of Pond Pollution Model

This paper specifically develops a computational approach in estimating the amount of pond pollution and determining the long time(every 48 hours) behavioural representation of the pond pollution model.This approach, which can be considered as an extension of the block predictor-corrector methods in form of implicit block multistep method has many computational advantages usingvariable step size technique. Moreover, it possesses some important advantages of designing a suitable step size, stopping criteria(prescribed tolerance level) and error control/minimization as well. This makes the new approach specifically efficient for solving systems of first-order ordinary differentialequations. Analysis of some theoretical properties of the method is carried out to ascertain the extent of performance of the method. Again, numerical results are given to display the performance and efficiency of this new method on system first-order ordinary differential equations

Block Algorithm for General Third Order Ordinary Differential Equation

We present a block algorithm for the general solution o

Four Steps Block Predictor-Block Corrector Method for the solution of y"=f(x,y,y)

A method of collocation and interpolation of the power series approximate solution at some selected grid points is considered to generate a continuous linear multistep method with constant step size.predictor-corrector method was adopted where the predictors and the correctors considered two and three interpolation points implemented in block method respectively.The efficiency of the proposed method was tested on some numerical examples and found to compete favorably with the existing methods